Temporal topological operators with picture fuzzy multifunctions

D. L. Shi; M. N. Abu_Shugair; S. E. Abbas; Ismail Ibedou; D. L. Shi; M. N. Abu_Shugair; S. E. Abbas; Ismail Ibedou

doi:10.3934/math.2025738

AIMS Mathematics

2025, Volume 10, Issue 7: 16477-16497. doi: 10.3934/math.2025738

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Temporal topological operators with picture fuzzy multifunctions

1.
College of Accounting, Guangzhou College of Technology and Business, Guangzhou 528138, China
2.
Mathematics Department, College of Science, Jazan University, Jazan 45142, Saudi Arabia
3.
Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt
4.
Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt

Received: 19 February 2025 Revised: 03 July 2025 Accepted: 09 July 2025 Published: 22 July 2025

This paper introduces the notion of a temporal picture fuzzy modal topological structure (TPFMTS). These structures are based on novel temporal picture fuzzy topological operators as common modal topological operators standing for the closure and interior operators. The paper discusses several fundamental properties of these new TPFMTSs. The constructed temporal picture fuzzy modal topological operators establish that some properties that were considered satisfactory in the temporal intuitionistic fuzzy modal topological structures, as defined by Atanassov in 2023, are not fulfilled. Also, we introduce the concepts of $ TPF^u $ and $ TPF^l $-continuous and almost continuous multifunctions. Several properties and characterizations of the presented continuous multifunctions and their types of continuity are established. Some examples are given to explain the correct implications between these notions.
- temporal picture fuzzy set,
- picture fuzzy modal operator,
- picture fuzzy topological structure,
- topology,
- ideal,
- multifunction
Citation: D. L. Shi, M. N. Abu_Shugair, S. E. Abbas, Ismail Ibedou. Temporal topological operators with picture fuzzy multifunctions[J]. AIMS Mathematics, 2025, 10(7): 16477-16497. doi: 10.3934/math.2025738

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Abstract

This paper introduces the notion of a temporal picture fuzzy modal topological structure (TPFMTS). These structures are based on novel temporal picture fuzzy topological operators as common modal topological operators standing for the closure and interior operators. The paper discusses several fundamental properties of these new TPFMTSs. The constructed temporal picture fuzzy modal topological operators establish that some properties that were considered satisfactory in the temporal intuitionistic fuzzy modal topological structures, as defined by Atanassov in 2023, are not fulfilled. Also, we introduce the concepts of $ TPF^u $ and $ TPF^l $-continuous and almost continuous multifunctions. Several properties and characterizations of the presented continuous multifunctions and their types of continuity are established. Some examples are given to explain the correct implications between these notions.

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